Hyperbolic two‑temperature generalized thermoelastic infinite medium with cylindrical cavity subjected to the non‑Gaussian laser beam

The present work is devoted to a study of the induced temperature and stress fields in an elastic infinite medium with a cylindrical cavity under the purview of hyperbolic two-temperature thermoelasticity. The medium is an isotropic homogeneous thermoelastic material. The bounding plane surface of the cavity is loaded thermally by a non-Gaussian laser beam with a pulse duration of two ps. An exact solution to the problem is obtained in the Laplace transform space and the inversions of the Laplace transform have been carried out numerically. The derived expressions are computed numerically for copper and the results are presented in graphical form. The two-temperature parameter has significant effects on all the studied functions and plays a vital role in the speed propagation of the thermomechanical waves.


Introduction
Although thermomechanical phenomena in most practical engineering applications are adequately simulated with the classical Fourier heat conduction equation, there is an important body of problems that require due consideration of thermomechanical coupling: it is appropriate in these cases to apply the generalized theory of thermoelasticity. Serious attention has been paid to the generalized thermoelasticity theories in solving thermoelastic problems in place of the classical uncoupled/coupled theory of thermoelasticity.
The absence of any elasticity term in the heat conduction equation for uncoupled thermoelasticity appears to be unrealistic since due to the mechanical loading of an elastic body, the strain so produced causes variation in the temperature field. Moreover, the parabolic type of the heat conduction equation results in an infinite velocity of thermal wave propagation, which also contradicts the actual physical phenomena. Introducing the strain-rate term in the uncoupled heat conduction equation, Biot extended the analysis to incorporate coupled thermoelasticity [1]. In this way, although the first shortcoming was over, there remained the parabolic type partial differential equation of heat conduction, which leads to the paradox of infinite 1 3 63 velocity of the thermal wave. To eliminate this paradox generalized thermoelasticity theory was developed subsequently. Due to the advancement of pulsed lasers, fast burst nuclear reactors and particle accelerators, etc. which can supply heat pulses with a very fast time-rise Bargmann [2] and Boley [3]; generalized thermoelasticity theory is receiving serious attention. The development of the second sound effect has been nicely reviewed by Chandrasekharaiah [4]. At present mainly two different models of generalized thermoelasticity are being extensively used-one proposed by Lord and Shulman [5] and the other proposed by Green and Lindsay [6]. L-S (Lord and Shulman theory) suggests one relaxation time and according to this theory, only Fourier's heat conduction equation is modified; while G-L (Green and Lindsay theory) suggests two relaxation times and both the energy equation, and the equation of motion are modified.
The so-called ultra-short lasers are those with pulse duration ranging from nanoseconds to femtoseconds in general. In the case of ultra-short-pulsed laser heating, the high-intensity energy flux and ultra-short duration laser beam, have introduced situations where very large thermal gradients or an ultra-high heating speed may exist on the boundaries Sun et al. [7]. In such cases, as pointed out by many investigators, the classical Fourier model, which leads to an infinite propagation speed of the thermal energy, is no longer valid Tzou, [8,9]. The non-Fourier effect of heat conduction considers the effect of mean free time (thermal relaxation time) in the energy carrier's collision process, which can eliminate this contradiction. Wang and Xu have studied the stress wave induced by nanoseconds, picoseconds, and femtoseconds laser pulses in a semi-infinite solid [10]. The solution considers the non-Fourier effect in heat conduction and the coupling effect between temperature and strain rate. It is known that characteristic elastic waveforms are generated when a pulsed laser irradiates a metal surface.
The two temperatures theory of thermoelasticity was introduced by Gurtin and Williams [11], Chen and Gurtin [12], and Chen et. al. [13,14], in which the classical Clausius-Duhem inequality was replaced by another one depending on two temperatures; the conductive temperature and the thermodynamic temperature T , the first is due to the thermal processes, and the second is due to the mechanical processes inherent between the particles and the layers of elastic material, this theory was also investigated by Ieşan [15].
The two-temperature model was underrated and unnoticed for many years thereafter. Only in the last decade has the theory been noticed, developed in many works, and found its applications mainly in the problems in which the discontinuities of stresses have no physical interpretations. Among the authors who contribute to developing this theory, Quintanilla studied existence, structural stability, convergence and spatial behaviour for this theory [16], Youssef introduced the generalized Fourier law to the field equations of the two-temperature theory of thermoelasticity and proved the uniqueness of solution for homogeneous isotropic material [17], Puri and Jordan studied the propagation of harmonic plane waves, recently [18], Magaña and Quintanilla [19] have studied the uniqueness and growth solutions for the model proposed by Youssef [17].
Youssef introduced another model of heat conduction law that depends on two unequal and different kinds of temperatures, dynamic temperature, and conducive temperature. The difference value between those types of temperatures is directly proportional to the value of the supplied heat [20]. Youssef noted that the thermal wave due to the two-temperature model propagates with infinite speed [21]. So, Youssef modified the two-temperature model to the hyperbolic two-temperature model in which the difference between the second derivative concerning the time of the dynamical and conductive temperature is proportional to the heat supply. He found that this model introduces a thermal wave that propagates with a limited speed [21].
The present work is devoted to a study of the induced temperature and stress fields in an elastic infinite medium with a cylindrical cavity under the purview of hyperbolic two-temperature thermoelasticity theory. The medium is an isotropic homogeneous thermoelastic material. The bounding plane surface of the cavity is loaded thermally by a non-Gaussian laser beam with a pulse duration of 2 ps. An exact solution to the problem is obtained in the Laplace transform space and the inversion of Laplace transforms has been carried numerically. The derived expressions are computed numerically for copper and the results are presented in graphical form.

The governing equations
Let us consider a perfectly conducting thermoelastic infinite body with a cylindrical cavity that occupies the region R ≤ r < ∞ of an isotropic homogeneous medium whose state can be expressed in terms of the space variable r and the time variable t such that all the field functions vanish at infinity. We use the cylindrical system of coordinates (r, , z) with the z-axis lying along the axis of the cylinder as in Fig. 1.
Due to symmetry, the problem is one-dimensional with all the functions considered depending on the radial distance r and the time t. It is assumed that no external forces are acting on the medium.
Thus, the field equations in the cylindrical one-dimensional case can be put as [20][21][22]: ij , i, j = r, , z are the components of the stress tensor, e is the cubic dilatation, u is the displacement,K is the thermal conductivity, o is the relaxation time, a is a non-negative parameter(two-temperature parameter), and Q is the heat source per unit mass.
A beam with initial temperature distribution T (x, z, 0) = T 0 will be considered. From time t = 0 its upper surface (z = 0) is irradiated uniformly by a laser pulse with a non-Gaussian temporal profile as [7]: where t p = 2 ps is a characteristic time of the laser pulse (the time duration of a laser pulse, see Fig. 2), L 0 is the laser intensity which is defined as the total energy carried by a laser pulse per unit area of the laser beam as in Fig. 1 [7].
The conduction heat transfer in the medium can be modelled as a one-dimensional problem with an energy source Q(t).
where is the absorption depth of heating energy and R a is the surface reflectivity. is the dimensionless two-temperature parameter, We apply the Laplace transform of both sides of the last equations defined as: Hence, we obtain: −̈=ã∇ 2 , To complete the solution in the Laplace, transform space, we will consider the medium described above is quiescent and the bounding plane of the cavity (r = R) has not any thermal or mechanical loading: After using Laplace transform, we have and After using Laplace transform, we have: Apply the last two conditions, we obtain: and After solving the last system of equations, we get: Finally, we have: To determine the conductive and thermal temperature, displacement and stress distributions in the time domain, the Riemann-sum approximation method is used to obtain the numerical results. In this method, any function in the Laplace domain can be inverted to the time domain as where R e is the real part and i is the imaginary number unit. For faster convergence, numerous numerical experiments have shown that the value satisfies the relation t ≈ 4.7 [9].

Numerical solutions and discussions
To illustrate the analytical procedure presented earlier, we now consider a numerical example for which computational results are given. For this purpose, copper is taken as the thermoelastic material for which we take the following values of the different physical constants [20]: (54) Fig. 3 The conductive temperature distribution Fig. 4 The thermo-dynamical temperature distribution Figure 3 shows the conductive temperature increment for the three studied models of thermoelasticity, and we can see that the values of the conductive temperature increment take the following order: It means that the two-temperature parameter has significant effects on the conductive temperature increment. Figure 4 shows the thermo-dynamical temperature increment for the three studied models of thermoelasticity, and we can see that the values of the conductive temperature increment take the following order: It means that the two-temperature parameter has significant effects on the thermo-dynamical temperature increment. Moreover, on the thermo-dynamical temperature increment, the effect of the two-temperature parameter is stronger than the conductive temperature increment. Figure 5 shows the radial stress for the three studied models of thermoelasticity, and we can see that the absolute values of the radial stress take the following order: It means that the two-temperature parameter has significant effects on the radial stress. Figure 6 shows the displacement for the three studied models of thermoelasticity, and we can see that the absolute values of the displacement take the following order: It means that the two-temperature parameter has significant effects on the displacement. Figure 7 shows the cubical deformation for the three studied models of thermoelasticity, and we can see that the absolute values of the cubical deformation take the following order: It means that the two-temperature parameter has significant effects on the cubical deformation.
According to the Figs. 3, 4, 5, 6, and 7, the two-temperature parameter plays a vital role in the speed of the thermos-mechanical waves propagation.

Conclusion
The goal of the current work was to investigate the induced temperature and stress fields in a hyperbolic two-temperature elastic infinite medium with a cylindrical hollow. Isotropic homogenous thermoelastic material makes up the medium; a non-Gaussian laser beam thermally loads the cavity's boundary plane surface. In Laplace transform space, a precise solution to the issue is discovered, and numerical Laplace transforms are completed. The obtained formulas are numerically calculated for copper, and the outcomes are displayed graphically. According to the results, we can conclude that: • The two-temperature parameter has significant effects on the conductive temperature increment. • The two-temperature parameter has significant effects on the thermo-dynamical temperature increment and the effect of the two-temperature parameter is stronger than the conductive temperature increment. • The two-temperature parameter has significant effects on the radial stress, displacement, and cubical deformation. • The two-temperature parameter plays a vital role in the speed propagation of the thermal and mechanical waves Authors' contributions AEB: constructed the model; wrote the introduction; reviewed the paper. HY: solved the application; figured out the results; reviewed the draft. MEN: wrote the discussions; wrote the conclusion; revised the English.
Funding Not Applicable.
Availability of data and materials Not Applicable.

Conflict of interest Not Applicable.
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